Variable stayout distance for beamhopping satellite

ABSTRACT

A system and method for scheduling a variable stayout distance when beam hopping, the method including providing an illumination area of a satellite and candidate beam centers disposed in the illumination area; measuring a respective scan angle from an antenna boresight to a respective beam center of the candidate beam centers; and determining a reuse factor k for each of the candidate beam centers, based on a proportion of the respective scan angle to a maximum scan angle. Each candidate beam center may be processed sequentially. Prior to adding each candidate beam center to a current beam center set, checking whether a candidate beam center meets the stayout distance criteria from all beam centers already in the beam center set.

CROSS-REFERENCE TO RELATED APPLICATIONS AND INCORPORATION BY REFERENCE

The present application is a continuation of U.S. application Ser. No.16/729,870, filed Dec. 30, 2019, which is incorporated herein byreference in its entirety.

FIELD

A scheduler that adapts its stayout distance to accommodate a reducedperformance due to scan loss in a Very High Throughput Satellite (VHTS)system using beam hopping and antennas with high scan distortion towardsthe edges of a coverage area.

BACKGROUND

Classical satellite systems use fixed beam laydowns over time, typicallyimplementing a fixed reuse pattern (e.g., 3-color reuse). Some systemshave implemented “beam-hopping”, a beam laydown that is not constantover time, with the concept of a “stayout distance”. No two cells arepermitted to be in the illuminated set if the distance between the cellcenters is less than this stayout distance. The stayout distance isdesigned to limit interference between cells, for example, Co-channelinterference (CCI). The use of fixed stayout distance is disadvantageousin systems where the beam characteristics are not constant over thecoverage area.

Beam hopping satellites require a beam hopping scheduling mechanism thatneeds to accommodate a spatially and temporally varying traffic pattern.Previous beam hopping systems accounted for these factors but not thedegraded performance over the coverage area caused by changes in theantenna performance over that coverage area.

A satellite antenna will typically produce the most compact beamstowards the antenna boresite and will produce degraded beams as theangle between the boresite and the beam center increases, an effectreferred to as scan loss. The area covered by the beams at larger anglesfrom the boresite (scanned beams) is larger than the area covered by thebeams at the boresite.

SUMMARY

This Summary is provided to introduce a selection of concepts in asimplified form that is further described below in the DetailedDescription. This Summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used to limit the scope of the claimed subject matter.

The present teachings improve VHTS design. The VHTS is a major buildingblock of the satellite consumer, aeronautical, defense, government,enterprise and international business areas. The present teachingsdisclose a variable stayout distance to accommodate the loss of beamperformance as a function of scan angle.

A system of one or more computers can be configured to performparticular operations or actions by virtue of having software, firmware,hardware, or a combination of them installed on the system that inoperation causes or cause the system to perform the actions. One or morecomputer programs can be configured to perform particular operations oractions by virtue of including instructions that, when executed by dataprocessing apparatus, cause the apparatus to perform the actions. Onegeneral aspect includes a non-transient computer-readable storage mediumhaving instructions embodied there onto implement a method forscheduling a variable stayout distance when beam hopping. The methodincluding providing an illumination area of a satellite and candidatebeam centers disposed in the illumination area; measuring a respectivescan angle from an antenna boresight to a respective beam center of thecandidate beam centers; and determining a reuse factor k, for each ofthe candidate beam centers, based on a proportion of the respective scanangle to a maximum scan angle. Other embodiments of this aspect includecorresponding computer systems, apparatus, and computer programsrecorded on one or more computer storage devices, each configured toperform the actions of the methods.

In the following the terms “stayout distance” and “reuse factor” will beused. As is well known from the cellular radio field, for a fixed colorreuse pattern there is a relationship between a reuse factor k, adistance D between the centers of cells of the same color and a cellradius R, namely, D=√{square root over (3k)}R, where k is the number ofdistinct sets of orthogonal resources (colors).

Implementations may include one or more of the following features. Themethod where the illumination area includes imaginary cells superimposedon the illumination area, each cell has a cell center, and each of thecandidate beam centers includes one of the cell centers. The imaginarycells are substantially hexagonal in shape. The method where the reusefactor k for each of the candidate beam centers is constrained by thehexagonal geometry. The method where the centers of the imaginary cellsare not restricted to a hexagonal lattice. The method may includesetting the reuse factor k for each of the candidate beam centers bychoosing either a next smallest reuse factor k1 or a next largest reusefactor k2 from a set of reuse factors based on a probability p. Themethod may include generating a current beam center set by sequentiallyadding a respective candidate beam center of the candidate beam centerswhen the respective candidate beam center is outside a respective reusedistance D from each of the beam centers already in the current beamcenter set. The candidate beam centers are ordered by a traffic metricassociated with each of the candidate beam centers. Implementations ofthe described techniques may include hardware, a method or process, orcomputer software on a computer-accessible medium.

One general aspect includes a beam forming system to schedule using avariable stayout distance when beam hopping. The system includes asatellite covering an illumination area and candidate beam centersdisposed in the illumination area; and a stayout scheduler to measure arespective scan angle from an antenna boresight to a respective beamcenter of the candidate beam centers, and to determine a reuse factor k,for each of the candidate beam centers, based on a proportion of therespective scan angle to a maximum scan angle. Other embodiments of thisaspect include corresponding computer systems, apparatus, and computerprograms recorded on one or more computer storage devices, eachconfigured to perform the actions of the methods.

Additional features will be set forth in the description that follows,and in part will be apparent from the description, or may be learned bypractice of what is described.

DRAWINGS

In order to describe the manner in which the above-recited and otheradvantages and features may be obtained, a more particular descriptionis provided below and will be rendered by reference to specificembodiments thereof which are illustrated in the appended drawings.Understanding that these drawings depict only typical embodiments andare not, therefore, to be limiting of its scope, implementations will bedescribed and explained with additional specificity and detail with theaccompanying drawings.

FIG. 1 illustrates a beam forming system to schedule beam hopping usinga variable stayout distance according to various embodiments.

FIG. 2 illustrates a method for scheduling a variable stayout distancewhen beam hopping according to various embodiments.

FIG. 3 illustrates an example hexagonal grid showing cellular reuse k=3(i=1, j=1), according to various embodiments.

FIG. 4 illustrates an example hexagonal grid showing cellular reuse k=4(i=0, j=2), according to various embodiments.

Throughout the drawings and the detailed description, unless otherwisedescribed, the same drawing reference numerals will be understood torefer to the same elements, features, and structures. The relative sizeand depiction of these elements may be exaggerated for clarity,illustration, and convenience.

DETAILED DESCRIPTION

The present teachings may be a system, a method, and/or a computerprogram product at any possible technical detail level of integration.The computer program product may include a computer readable storagemedium (or media) having computer readable program instructions thereonfor causing a processor to carry out aspects of the present invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, or either source code or object code written in anycombination of one or more programming languages, including an objectoriented programming language such as SMALLTALK, C++ or the like, andconventional procedural programming languages, such as the “C”programming language or similar programming languages. The computerreadable program instructions may execute entirely on the user'scomputer, partly on the user's computer, as a stand-alone softwarepackage, partly on the user's computer and partly on a remote computeror entirely on the remote computer or server. In the latter scenario,the remote computer may be connected to the user's computer through anytype of network, including a local area network (LAN) or a wide areanetwork (WAN), or the connection may be made to an external computer(for example, through the Internet using an Internet Service Provider).In some embodiments, electronic circuitry including, for example,programmable logic circuitry, field-programmable gate arrays (FPGA), orprogrammable logic arrays (PLA) may execute the computer readableprogram instructions by utilizing state information of the computerreadable program instructions to personalize the electronic circuitry,in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the block may occur out of theorder noted in the figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

Reference in the specification to “one embodiment” or “an embodiment” ofthe present invention, as well as other variations thereof, means that afeature, structure, characteristic, and so forth described in connectionwith the embodiment is included in at least one embodiment of thepresent invention. Thus, the appearances of the phrase “in oneembodiment” or “in an embodiment”, as well any other variations,appearing in various places throughout the specification are notnecessarily all referring to the same embodiment.

In the current teachings, a distance measure refers to U, V coordinates.U, V coordinates are angles, measured from a satellite's antenna(boresite) point of view. As such, in the present teachings the term“distance” is an angular distance. Moreover, in the present teachings, ause of the well-known concept of reuse color is not advocated. The reusecolor concept is being used to find the set of possible distancesbetween beam centers.

Classical satellite communication systems have typically implemented afixed cellular reuse to control inter-cell interference. Cells havingcell centers that are co-incident with a beam center 120 are laid ontothe earth's surface to provide an illumination area 110 of the satellite106. Signal levels from a beam directed to a target cell are typicallyhigh enough to cause significant interference to an immediately adjacentcell to the target cell. As such, illumination of the immediatelyadjacent cells may be chosen to use orthogonal resources to limitinterference. Orthogonal resources could be frequency, time, and/orpolarization. For example, in a 3-color reuse design there might be 3different (orthogonal) frequency bands. The cells are colored (e.g., R,G, B) and adjacent cells are assigned colors (frequency bands) so thatno two immediately adjacent cells have the same color.

Some more recent satellite communication systems have implementedsystems in which the beam laydowns are not constant, also referred to as“beam hopping” systems without a concept of a variable stayout distance.In this beam hopping system, the set of cells sharing non-orthogonalresources (e.g., operating at the same time, frequency, andpolarization) is not constant; however, the stayout distance in thisprior art system is constant.

The principle of having a variable stayout distance between beam centerscan be applied to a general beamforming system, for example, abeamforming system aiming beam centers at arbitrary points. In this casethe stayout distance D would not be restricted to distances matching thedistance between fixed beam centers on a hexagonal grid.

In some embodiments, the beamforming system may be simplified by using aset of potential beam centers that are not arbitrary points but arerestricted to a certain set of potential points. In some systems, ahexagonal grid tiles the plane and the centers of this grid are the setof possible beam centers. Hexagonal grid lines are used in radiocommunication systems, such as cellular systems (with or without beamforming), satellite systems, or the like. When restricting the distancebetween active beam centers on this grid to be greater than some minimalvalue, the distances between potential beam centers can only take oncertain discrete values.

When beam centers are restricted to cell centers of a predefined grid(for example, the hexagonal grids of FIG. 3 and FIG. 4 ), an averagestayout distance may be approximately achieved. Arbitrary stayoutdistances are not always achievable and are thus approximated with adiscrete set of distances and the “coin toss”. When implementing anaverage stayout, for each beam center, the process decides between thenext largest stayout (compared to the desired average stayout) and thenext smallest stayout (compared to the desired average stayout) bychoosing between them with some probability. One obvious extension wouldbe the use of three or more stayout distances.

FIG. 1 illustrates a beam forming system to schedule beam hopping usinga variable stayout distance according to various embodiments.

FIG. 1 shows an example of a beam forming system 100 to schedule using avariable stayout distance when beam hopping. In FIG. 1 , a beam center120 is marked with an X, and a beam definition 122 is illustrated as agreyed region centered on the beam center 120. It is seen that the beamsnear the boresite (near the center of coverage area, i.e., at the origin0,0) have a better beam definition (less diffused) than the beamsfarther from the boresite (near the corners of the coverage area, i.e.,at −1, 3). The better-defined beams near the center cover smaller areas(in and around the targeted cell) as compared to the beams farther fromthe boresight. Therefore, the beams at the large scan angle are moresubject to interference from adjacent beams. It would be advantageousfor an overall system throughput to employ a larger stayout distance forthe beams at the large scan angle and a smaller stayout distance forbeams with small scan angle. A stayout scheduler 122 schedules beamhopping using a variable stayout distance.

A useful guide to determining a variable stayout is to employ a reusefactor k1, corresponding to scan angle S1 (see FIG. 1 ) near theboresite and a reuse factor k2, corresponding to a maximum scan angleS2. For a beam at a scan angle S, the reuse factor k may be defined ask=F(S) for some function F. The function F could be any function chosento vary the stayout distance according to scan angle. For example,suppose the minimum k is k_(min), maximum k is k_(max), and the maximumscan angle is S_(max), the reuse factor k may be calculated as:

$k = {{F(S)} = {{k\;\min \times \left( {1 - \left( \frac{S}{S\;\max} \right)^{\alpha}} \right)} + {k\;\max \times \left( \frac{S}{S\;\max} \right)^{\alpha}}}}$The exponent α might be set to 2 for example.

In the exemplary system, after determining or choosing the variablereuse factor k the variable reuse may be implemented. One way toimplement the variable reuse is to find the corresponding reuse distanceD by applying the previous formula relating D to k. This approach wouldbe ideal for systems in which the cell centers are not restricted to lieon a hexagonal grid in which case the reuse distance D is applieddirectly as the stayout distance associated with that cell center. Forsystems where the locations of the possible beam centers form ahexagonal grid, the distances between beam centers cannot take on allpossible values. So, for an arbitrary stayout distance D, a minimalinter-beam distance will be some other D′, where D′ corresponds to thenext possible reuse distance greater than D. Because of this, a betterapproach to determine the stayout distance may be implemented as a “cointoss”.

Suppose we wish to implement a system where the inter-beam distancescorrespond to a reuse factor having an average value of k. As is wellknown, not all values of reuse factor k are possible; only k=i²+ij+j²for integers i, j are possible.

A “coin toss” can choose between the next lowest reuse distance k1<k andthe next higher k2>k. Then for any potential beam, a stayout distancemay be chosen by flipping a biased coin and choosing a distancecorresponding to k2 with probability p and choosing a distancecorresponding to k1 with probability 1−p, where an exemplary p may bechosen by

$p = \frac{k - {k1}}{{k2} - {k1}}$

Using this scheme, the average reuse factor for the system will be k asis easily verified. Combining the variable reuse described above forchoosing the reuse factor as a function of a cell's scan angle with thecoin toss scheme for implementing the variable reuse results in a designwhere the average reuse factor varies as a function of scan angle andthe reuse choses locations corresponding to cell centers on thehexagonal grid.

FIG. 2 illustrates a method for scheduling a variable stayout distancewhen beam hopping according to various embodiments.

In one embodiment, a beam-forming beam-hopping system may implement amethod 200 for scheduling a variable stayout distance when beam hopping.The system may be provided candidate beam centers per operation 202.Beam centers included as candidate beam centers may be changed and/orreordered at each time step (epoch). In some embodiments, the set ofbeam centers at each time step may be different and determined by, forexample, traffic demand, fade at cell. Scan angles from an antennaboresight to candidate beam centers may be measured per operation 204.Here, measuring a scan angle includes obtaining the scan angle from atable and the like. Each candidate beam center may be processedsequentially to determine a reuse factor k for each of the candidatebeam centers, based on a proportion of the respective scan angle to amaximum scan angle per operation 206. The candidate beam centers may beordered by a traffic metric per operation 210, for example, by highestto lowest traffic demand, traffic age, traffic priority, traffic Qualityof Service guarantee, or the like. Per operation 212, when generating acurrent beam center set by adding each candidate beam center, operation212 checks whether a candidate beam center meets a respective reusedistance D from each of the candidate beam centers already in thecurrent beam center set.

In some embodiments, each beam center may have a stayout distanceassociated with it. In some embodiments, the stayout distance isrecomputed each epoch, for example, to account for the vagaries of the“coin toss.” In other embodiments, while adding a beam center to thebeam center set, the process may generate a stayout distance criteriafor that beam center via a (pseudo)-random process, so that on average adesired value of stayout distance is produced. For example, the process200 may set the reuse factor k for each of the candidate beam centers bychoosing either a next smallest reuse factor k1 or a next largest reusefactor k2 based on a probability p per operation 208.

FIG. 3 illustrates an example hexagonal grid showing cellular reuse k=3(i=1, j=1), according to various embodiments.

FIG. 4 illustrates an example hexagonal grid showing cellular reuse k=4(i=0, j=2), according to various embodiments.

In the case of reuse 3 (FIG. 3 ), 3 colors illuminate the grid. A k=3can be provided by setting i=1 and j=1 (k=i²+ij+j²) In the case of reuse4 (FIG. 4 ), there are 4 colors. A k=4 can be provided by setting i=0and j=2. The closest cell centers that can be simultaneously illuminated(same time/frequency/polarization) are the ones that are shown as thesame color. In the case of k=3 the closest same color cell centers areat distance 3R, while for k=4 the distance is 2√{square root over (3)}R,where R is the radius of the hexagon.

As an example, a system level computer simulation was conducted toillustrate the benefit of the variable stayout concept. This simulationwas for a satellite system covering a large number of users across thecontinental US. In this example a hexagonal grid was used for thepotential beam centers, as described herein. Two cases are compared withthe only difference between them being that one case has a fixed reusefactor k=3 while the other has variable reuse factor over the range k=[3. . . 5]. The fixed reuse system delivers 3.73 units of throughput,while the variable reuse delivers 3.84 units.

Having described preferred embodiments of a system and method (which areintended to be illustrative and not limiting), it is noted thatmodifications and variations can be made by persons skilled in the artconsidering the above teachings. It is therefore to be understood thatchanges may be made in the embodiments disclosed which are within thescope of the invention as outlined by the appended claims. Having thusdescribed aspects of the invention, with the details and particularityrequired by the patent laws, what is claimed and desired protected byLetters Patent is set forth in the appended claims.

We claim as our invention:
 1. A non-transient computer-readable storagemedium having instructions embodied thereon, the instructions beingexecutable by one or more processors to perform a method for schedulinga variable stayout distance when beam hopping, the method comprising:providing an illumination area of a satellite and candidate beam centersdisposed in the illumination area; measuring a respective scan anglefrom an antenna boresight to a respective beam center of the candidatebeam centers; determining a reuse factor k, for each of the candidatebeam centers, based on a proportion of the respective scan angle to amaximum scan angle; and adapting the variable stayout distance of astayout schedular based on the reuse factor k.
 2. The method of claim 1,wherein the illumination area comprises imaginary cells superimposed onthe illumination area, each cell has a cell center, and wherein each ofthe candidate beam centers comprises one of the cell centers.
 3. Themethod of claim 2, wherein the centers of the imaginary cells arerestricted to a hexagonal lattice.
 4. The method of claim 2 wherein thecenters of the imaginary cells are not restricted to a hexagonallattice.
 5. The method of claim 2, wherein the imaginary cells aresubstantially hexagonal in shape.
 6. The method of claim 1, furthercomprising setting the reuse factor k for each of the candidate beamcenters by choosing either a next smallest reuse factor k1 or a nextlargest reuse factor k2 from a set of reuse factors based on aprobability p.
 7. The method of claim 1, wherein the reuse factor k foreach of the candidate beam centers is calculated as$k = {{F(S)} = {{k\;\min \times \left( {1 - \left( \frac{S}{S\;\max} \right)^{\alpha}} \right)} + {k\;\max \times \left( \frac{S}{S\;\max} \right)^{\alpha}}}}$with α being
 2. 8. The method of claim 1, further comprising generatinga current beam center set by sequentially adding a respective candidatebeam center of the candidate beam centers when the respective candidatebeam center is outside a respective reuse distance D from each of thecandidate beam centers already in the current beam center set.
 9. Themethod of claim 8, wherein the candidate beam centers are ordered by atraffic metric associated with each of the candidate beam centers. 10.The method of claim 8, further comprising setting the reuse factor k foreach of the candidate beam centers by choosing either a next smalleststayout distance k1 or a next largest reuse factor k2 from a set ofstayout distances based on a probability p, wherein the illuminationarea comprises substantially hexagonal imaginary cells superimposed onthe illumination area, each cell has a cell center, and each of thecandidate beam centers comprises one of the cell centers, and whereinthe reuse factor k for each of the candidate beam centers is calculatedas$k = {{F(S)} = {{k\;\min \times \left( {1 - \left( \frac{S}{S\;\max} \right)^{\alpha}} \right)} + {k\;\max \times \left( \frac{S}{S\;\max} \right)^{\alpha}}}}$with α being
 2. 11. A beam forming system to schedule using a variablestayout distance when beam hopping, the system comprising: a satellitecovering an illumination area and candidate beam centers disposed in theillumination area; and a stayout scheduler to measure a respective scanangle from an antenna boresight to a respective beam center of thecandidate beam centers, to determine a reuse factor k, for each of thecandidate beam centers, based on a proportion of the respective scanangle to a maximum scan angle, and to adapt the variable stayoutdistance of the stayout schedular based on the reuse factor k.
 12. Thesystem of claim 11, wherein the illumination area comprises imaginarycells superimposed on the illumination area, each cell has a cellcenter, and wherein each of the candidate beam centers comprises one ofthe cell centers.
 13. The system of claim 12, wherein the centers of theimaginary cells are restricted to a hexagonal lattice.
 14. The system ofclaim 12, wherein the centers of the imaginary cells are not restrictedto a hexagonal lattice.
 15. The system of claim 12, wherein theimaginary cells are substantially hexagonal in shape.
 16. The system ofclaim 11, wherein the stayout scheduler sets the reuse factor k for eachof the candidate beam centers by choosing either a next smallest reusefactor k1 or a next largest reuse factor k2 from a set of stayoutdistances based on a probability p.
 17. The system of claim 11, whereinthe reuse factor k for each of the candidate beam centers is calculatedas$k = {{F(S)} = {{k\;\min \times \left( {1 - \left( \frac{S}{S\;\max} \right)^{\alpha}} \right)} + {k\;\max \times \left( \frac{S}{S\;\max} \right)^{\alpha}}}}$with α being
 2. 18. The system of claim 11, wherein the stayoutscheduler generates a current beam center set by sequentially adding arespective candidate beam center of the candidate beam centers when therespective candidate beam center is outside a respective reuse distanceD from each of the candidate beam centers already in the current beamcenter set.
 19. The system of claim 18, wherein the candidate beamcenters are ordered by a traffic metric associated with each of thecandidate beam centers.
 20. The system of claim 18, wherein the stayoutscheduler sets the reuse factor k for each of the candidate beam centersby choosing either a next smallest reuse factor k1 or a next largestreuse factor k2 from a set of stayout distances based on a probabilityp, wherein the illumination area comprises substantially hexagonalimaginary cells superimposed on the illumination area, each cell has acell center, and each of the candidate beam centers comprises one of thecell centers, and wherein the reuse factor k for each of the candidatebeam centers is calculated as$k = {{F(S)} = {{k\;\min \times \left( {1 - \left( \frac{S}{S\;\max} \right)^{\alpha}} \right)} + {k\;\max \times \left( \frac{S}{S\;\max} \right)^{\alpha}}}}$with α being 2.